I can’t be sure of helping with the proof, though, until you tell us what $P(n;i_1,i_2,\dots,i_k)$ is. I’m going to guess that it’s the number of distinguishable permutations of a set of $n$ objects of $k$ types, $i_j$ being the number of indistinguishable objects of type $j$ for $j=1,\dots,k$. That makes your theorem a special case of the multinomial theorem.

Count the functions from $\{1,\dots,n\}$ to $\{1,\dots,k\}$ in two ways. You can think of such a function as an assigment of labels $1,\dots,k$ to the integers $1,\dots,n$. The term $P(n;i_1,\dots,i_k)$ is the number of ways to assign $i_1$ labels $1$, $i_2$ labels $2$, ..., and $i_k$ labels $k$.